One Step of Gradient Descent is Provably the Optimal In-Context Learner with One Layer of Linear Self-Attention

• Published in: arXiv.org
• Main Authors: Mahankali, Arvind, Hashimoto, Tatsunori B, Ma, Tengyu
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 07.07.2023
• Subjects: Algorithms; Context; Least squares; Normal distribution; Regression; Regression analysis; Statistical analysis; Training; Transformers
• ISSN: 2331-8422

Recent works have empirically analyzed in-context learning and shown that transformers trained on synthetic linear regression tasks can learn to implement ridge regression, which is the Bayes-optimal predictor, given sufficient capacity [Aky\"urek et al., 2023], while one-layer transformers with linear self-attention and no MLP layer will learn to implement one step of gradient descent (GD) on a least-squares linear regression objective [von Oswald et al., 2022]. However, the theory behind these observations remains poorly understood. We theoretically study transformers with a single layer of linear self-attention, trained on synthetic noisy linear regression data. First, we mathematically show that when the covariates are drawn from a standard Gaussian distribution, the one-layer transformer which minimizes the pre-training loss will implement a single step of GD on the least-squares linear regression objective. Then, we find that changing the distribution of the covariates and weight vector to a non-isotropic Gaussian distribution has a strong impact on the learned algorithm: the global minimizer of the pre-training loss now implements a single step of \(\textit{pre-conditioned}\) GD. However, if only the distribution of the responses is changed, then this does not have a large effect on the learned algorithm: even when the response comes from a more general family of \(\textit{nonlinear}\) functions, the global minimizer of the pre-training loss still implements a single step of GD on a least-squares linear regression objective.